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Abstract
High peak-to-average power ratio (PAPR) of the signal is a major drawback in optical orthogonal frequency division multiplexing (OFDM) system. In this paper, an intensity-modulated type Partial Transmit Sequences (PTS) based scheme is proposed and applied to the intensity-modulated OFDM (IMDD-OFDM) system. The proposed intensity-modulated type PTS (IM-PTS) scheme ensures that the time-domain signal output by the algorithm is real value. What’s more, the complexity of the IM-PTS scheme has been reduced without much performance penalty. A simulation is performed to compare the PAPR of different signal. In the simulation, the PAPR of OFDM signal is reduced from 14.5 dB to 9.4 dB at 10−4 probability. We also compare the simulation results with another algorithm based on the PTS principle. A transmission experiment is conducted in a seven-core fiber IMDD-OFDM system at a rate of 100.8Gbit/s. The Error Vector Magnitude (EVM) of received signal is reduced from 9 to 8 at -9.4dBm received optical power. Furthermore, the experiment result shows that the reduction of complexity has little performance impact. The optimized intensity-modulated type PTS (O-IM-PTS) scheme effectively increases the tolerance of the nonlinear effect of the optical fiber and reduces the requirement for linear operating range of optical device in the transmission system. During the upgrade process of the access network, there is no need to replace the optical device in the communication system. What’s more, the complexity of PTS algorithm has been reduced, which lower data processing performance requirements of the devices such as ONU and OLT. As a result, the cost of network upgrades is reduced a lot.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Orthogonal frequency division multiplexing (OFDM) is an attractive technique for high-speed transmission, which disperses the high-speed original data to each mutually orthogonal low-speed sub-carrier through serial-parallel transformation [1]. The technology has excellent performance in complex channel interference environments. When OFDM is applied to optical fiber communication, it can improve transmission capacity while effectively addressing dispersion in the fiber and reducing network upgrade costs [2,3]. However, the inherent disadvantages of OFDM technology have also been brought into optical fiber communication [2]. One of the main problems: the high Peak to Average Power Ratio (PAPR), which is caused by OFDM technology using multiple subcarriers overlapped to generate the transmit signal [4]. A high PAPR can cause some time-domain signals to be outside the linear operating range of the hardware, causing distortion and hardware wear [5,6].
Several techniques are available to reduce the PAPR of OFDM signals. Amplitude limiting class technique is simple, but can seriously affect the signal BER [7,8]. The coding technique needs a large lookup table at sender and receiver, which consumes large amounts of data storage and computation [9,10]. The scrambling technique is easier to decode at the receiver, and it does not cause error codes, but it also has the problem of high computational complexity at the sender [11,12].
Partial transmit sequences (PTS) is a kind of scrambling technique, which originally proposed by Stefan H. Müller and Johannes B. Huber in 1997 [13]. The basic flow is shown in Fig. 1. At a symbol of OFDM signal, subcarrier data is partitioned into a number of disjoint sub-blocks, and zero-filled to generate sequences of sub-blocks, which have equal length to the original OFDM symbol. After the inverse discrete Fourier transform (IDFT), sequences of sub-blocks are converted into time-domain sequences, and the phase optimization gives a set of rotation factor combination for the time-domain sequences of each sub-blocks. Finally, the time-domain sequence of each sub-block after applying the rotation factor is summed to obtain the lowest PAPR transmit signal, and the rotation factors are sent as sideband information.
There are currently a lot of improvements applied to the PTS algorithm. Literature [14] uses μ-law to compress and expand the optimization results of the PTS algorithm, thus improves the effectiveness of the algorithm. But the literature does not propose a solution to reduce the complexity of PTS algorithm. Literature [15] focuses on the application of PTS algorithm with OFDM technology in the optical carrier wireless communication. The performance of the three kinds of PTS algorithms is compared experimentally in this literature. But the literature does not propose a solution to reduce the complexity of PTS algorithm. Literature [16] enhances the search results of the PTS algorithm using the particle swarm algorithm, thus indirectly reducing the complexity of the PTS algorithm. But the literature does not perform a transmission test. Literature [17] focuses on the combination of PTS algorithm and chaos-based encryption and a transmission experiment was conducted. But the literature increases the complexity of PTS algorithm due to the encryption. Literature [18] also introduces chaos-based encryption into PAPR reduction. Combining SLM and PTS, literature achieves good security while reducing PAPR. However, the literature does not propose a solution to reduce the complexity. Literature [19] optimizes traditional PTS with improved flower pollination (IFT) algorithm. The proposed IFT-PTS algorithm is applied to the asymmetrically clipped optical OFDM (ACO-OFDM) system. A balance has been achieved in performance and complexity, but the experimental results in the literature are limited to simulation, and no actual experiments have been carried out. What’s more, the proposed improvement does not consider the characteristics of the OFDM system, which may be unstable performance. In addition, PTS algorithm was originally proposed to optimize coherent communication systems. When applied to low-cost intensity-modulated communication systems, unconstrained sub-block partition and randomized rotation factor cannot guarantee that the output signal is real value.
In this paper, we propose the IM-PTS scheme. This scheme can guarantee that the output OFDM signal is real value under any sub-block partition and any rotation angle by applying appropriate constraints. Base on the IM-PTS scheme, we try to reduce complexity of PTS algorithm from a total of four perspectives: limiting the rotation angle, using equivalent operations, removing duplicate combinations, and using local signal optimization to fit the global signal optimization. Compare to the original PTS scheme, the proposed IM-PTS and O-IM-PTS scheme can be directly applied to IMDD-OFDM system. Also, the complexity of PTS algorithm is reduced a lot with little performance loss at O-IM-PTS scheme.
Simulation results show that after reducing a lot of complexity, the O-IM-PTS scheme can basically reach the performance of the IM-PTS scheme. Additional simulation has been done in order to compare the proposed scheme with literature [18], showing that the algorithm proposed in this paper has a greater advantage in terms of computational complexity while achieving approximate results. We also conducted experiments in a seven-core fiber optic to compare the quality of different receiving signal. The experimental results confirm that the PAPR reduction of the OFDM signal by any PTS algorithm effectively improves the signal quality at the receiver. What’s more, the difference of IM-PTS scheme and O-IM-PTS scheme can be considered as experimental deviation.
2. Principle
The Fig. 2 shows the scheme of IM-PTS algorithm. Compared with the original PTS algorithm, this IM-PTS algorithm flow is as follows.
- 1. The input constellation data ${X_z}[k ]$ are mapped into a sequence $X[k ]$ of length $2N$ by performing Hermitian symmetry after S/P conversion. The mapping relationship is as Eq. (1):
In Eq. (1), denotes the real part of the sequence x; $imag(x )$ denotes the imaginary part of the sequence x; and denotes the complex conjugate sequence of the sequence.
- 2. The frequency domain sequence $X[k ]$ is sub-block partitioned by different areas. The area with $k = 0,N$ is the unique sub-block area, the area with $1 \le k \le N - 1$ is the master sub-block area, and the area with $N + 1 \le k \le 2N - 1$ is the slave sub-block area. The characteristics are as follows:
The unique sub-block area is partitioned into a separate sub-block. In most cases the sub-block is 0 and no processing is needed. When the length of ${X_{raw}}[k ]$ is N, the sub-block data is not 0, the sub-block is valid and is recorded as ${X_0}[k ]$ and processed separately. The master sub-block area can be partitioned using any kind of methods, commonly used are adjacent partitioning, interleaved partitioning, pseudo-random partitioning. The partitioned sub-block is called the master sub-block. Let a total of M disjoint master sub-blocks be partitioned. Each sub-block is expanded to a sequence of length $2N$ by zero-filling. The set of elements selected by the $m$th master sub-block is denoted as $\{{{X_m}} \}$, and the sequence is denoted as ${X_m}[k ]$, $1 \le m \le M$.
The partition result of the slave sub-block area depends entirely on the partition result of the master sub-block, which is also partitioned into M disjoint sub-blocks. Each sub-block is zero-filled to expand into a $2N$ length sequence. The set of elements chosen by the $m$th slave sub-block is denoted as $\{{X_m^\ast } \}$, and the sequence is denoted as $X_m^\ast [k ]$, $1 \le m \le M$. It needs to satisfy that if element $X[k ]\in \{{{X_m}} \}$, then there must be element $X[{2N - k} ]\in \{{X_m^\ast } \}$, $1 \le k \le N - 1$; conversely if the element $X[k ]\notin \{{{X_m}} \}$, then there must be element $X[{2N - k} ]\notin \{{X_m^\ast } \}$, $1 \le k \le N - 1$.
- 3. Perform $2N$-point IDFT on all sequences ${X_m}[k ]$. The resulting time-domain sequences are $x_m^\ast [n ]$ and $0 \le n \le 2N - 1$. Then add the unique sub-block (if it exists) for phase optimization. After optimization, the rotation factor for $m$th master sub-block is written as ${b_m}$ and the rotation factor for $m$th slave sub-block is written as $b_m^\ast $ and there is $b_m^\ast = conj({{b_m}} )$. The rotation factor for unique sub-block is written as ${b_0}$. All the rotation factors are sent as the sideband information.
- 4. If the unique sub-block exists, a $2N$-point IDFT is performed after applying the phase factor ${b_0}$ to convert the unique sub-block into a time-domain sequence, denoted as ${x_0}[n ]$. The formula is as Eq. (2):
- 5. Multiply each rotation factor obtained from phase optimization with each time-domain sequence. Then sum all time-domain sequences (including unique sub-block, if exists) to get the optimized OFDM signal $x[n ]$. The formula is as Eq. (3)
Theorem:
The IM-PTS scheme ensures that the output time-domain signal x[n] is real at any rotation factor.
Proof:
Let the element $X[a ]$, $1 \le a \le N - 1$ of the sequence $X[k ]$ belongs to a master sub-block set $\{{{X_m}} \}$, $1 \le m \le M$, i.e., $X[a ]\in \{{{X_m}} \}$, then there must be $X[{2N - a} ]\in \{{X_m^\ast } \}$.
$X[a ]$ and $X[{2N - a} ]$ are extended to two $2N$ length sequences of only one non-zero element by zero-filling. IDFT is performed to obtain time-domain sequences, denoted as ${x_a}[n ]$ and $x_a^\ast [n ]$. According to the definition of IDFT [20], the two time-domain sequences can be expressed as Eqs. (4)–(6):
In Eq. (4), Eq. (5), ${W_N}$ is called the rotation factor [20] in the DFT, which is defined as Eq. (7):
According to the periodicity of the rotation factor and the mapping of Hermitian symmetry in Eq. (1), the two time-domain sequences in Eq. (4), Eq. (5) can be reduced to Eq. (8), Eq. (9):
Assuming that the PTS-optimized rotation factors applied to the two subblocks in Eq. (8), Eq. (9) are ${b_a}$ and $b_a^\ast $, the optimization process can be expressed as Eq. (10), Eq. (11):
According to the multiplication principle of conjugate symmetry, . Since,, so we can reach Eq. (12):
Equation (12) shows that, when ${x_a}[n ]\cdot {b_a}$ and $x_a^\ast [n ]\cdot b_a^\ast $ are added, their imaginary parts cancel each other out.
Finally, the time domain signal of unique sub-block in Eq. (2) after applying rotation factor ${b_0}$ can be expressed as Eq. (13):
According to the definition of rotation factor in Eq. (7), we have, thus Eq. (13) can be expressed as Eq. (14):
Equation (14) guarantees that the unique sub-block remains a sequence of real value after the PTS algorithm.
In summary, the IM-PTS scheme ensures that the output time domain sequence is a real-value signal.
The flow chart of an intensity-modulated OFDM system after applying IM-PTS algorithm is shown in Fig. 3. After the binary data is generated, the constellation data is generated by QAM mapping. The ½ length DFT constellation data is generated by zero-filling. The IM-PTS algorithm (which contains the IDFT) generates the optimized OFDM symbols and sideband information from the constellation data. The cyclic prefix is added to each OFDM symbols to reduce the inter-symbol interference. After transmitted in the channel, the cyclic prefix is removed and DFT is processed. The PTS-decoding module uses sideband information to restore the original constellation data. Finally, the binary data is decoded from constellation data by QAM de-mapping.
However, for an optimum search algorithm, an unlimited search range would make the algorithm infeasible. In the IM-PTS algorithm the search range is the number of sub-blocks and the possible value of the rotation factors, which determine the complexity of the algorithm. Both need to be set as discrete values for the feasibility of the algorithm. Assuming that the rotation factor has P discrete values, the algorithm has M master sub-blocks, 0 unique sub-block, and $2N$-point DFT is used. When optimizing a single OFDM symbol, generating all sub-block data requires the computation of $2M$ times $2N$-point IDFT, the total number of possible rotation combinations is ${P^M}$, generating all candidate sequences requires ${P^M} \times 2N \times 2M$ times complex multiplication and ${P^M}$ times PAPR calculation. The complexity of this algorithm increases exponentially with the growth of the parameters, which is a heavy burden for the sender.
It is first necessary to limit the number of sub-blocks in the algorithm and the number of the rotation factor to make the algorithm feasible. We choose the set of rotation factors as $\{{1, - 1,j, - j} \}$, because the study [21] shows that this set of rotation factors can improve the efficiency of the system. The complex multiplications in Eq. (3) to generate candidate sequences in this case can be simplified. In order to let the DSP algorithm can be completed in a relatively short time, we choose 2,3,4,5 as the number of master sub-blocks. In addition, when the set of rotation factors is $\{{1, - 1,j, - j} \}$, half of possible combinations will be negative values of the existing combinations. The positive and negative signals have no effect on the calculation of power, and the PAPR of both generated signals is exactly the same. As a result, these combinations can be removed, reducing the possible combinations of rotation factors to ${P^M}/2$ species.
Secondly, according to Eq. (12), the time-domain sequence of the slave sub-blocks are conjugated symmetric with the time-domain sequences of the master sub-blocks, so the results of the slave sub-blocks can be directly generated from the results of the master sub-blocks. That is, computing $x_m^\ast [n ]\cdot b_m^\ast{=} conj({{x_m}[n ]\cdot {b_m}} )$ saves M times $2N$-point IDFT needed for the generation of slave sub-blocks. And the $2N \times M$ multiplications required to generate the candidate sequence from the slave sub-blocks can be simplified too.
In the original PTS scheme, the entire time-domain sequences of sub-blocks are involved in the optimization of PAPR, which is a heavy burden when the number of IDFT points is large. Literature [22] proposes an idea to reduce the complexity by selecting the time-domain signals with larger possible peak power for PAPR optimization. In other words, using the local optimization to fit the global optimization. The literature uses the Cauchy–Schwarz inequality to construct the parameter [22]:
And derive an inequality [22] to measure the maximum possible power of candidate time-domain signals at the $n$th time-domain sampling point in the case of M subblocks.
The inequality is for the original PTS algorithm, which is applicable to a wide range but not precise enough. For the IM-PTS algorithm proposed in this paper, we derive a new inequality to measure the maximum possible power.
The inequality is derived as follows:
Suppose the $n$th time-domain element of the $m$th master sub-block is ${x_m}[n ]= real({{x_m}[n ]} )+ imag({{x_m}[n ]} )\cdot j$. Then the $n$th time-domain element of the $m$th slave sub-block is $x_m^\ast [n ]= real({{x_m}[n ]} )- imag({{x_m}[n ]} )\cdot j$. Examine the effect of ${b_m},b_m^\ast $ on for 4 possible values of the rotation factor, with the following possibilities in Eq. (17):
Equation (17) can deduce to Eq. (18):
Using triangle inequality, Eq. (18) can derive to Eq. (19):
Taking the square on both sides of Eq. (19), we can reach Eq. (20):
This inequality in Eq. (20) is the measurement parameter used in this paper.
The parameter is optimized for intensity-modulated OFDM systems with rotation factors of $\{{1, - 1,j, - j} \}$, which has less complexity and more accuracy. In order to reflect the accuracy of the improved measurement parameter, a simulation is performed to compared the original parameter, the improved parameter, and the original signal power. Simulation parameters are: a total of 127 subcarriers, each subcarrier using 16-QAM modulation, DFT point is set as 256, the number of master sub-blocks is 4, using pseudo-random segmentation method, 4 times oversampling. Finally, select the first 100 time-domain sequences to compare the two measured parameters with the actual power. The simulation results are shown in Fig. 4. It can be seen that our measurement parameter is more accurate in measuring maximum possible power of the signal.
By calculating the maximum possible power, we can measure whether a certain time is likely to be the factor in the high PAPR of the signal. When generating the best rotation factor set, only the time domain signal with higher maximum possible power participates in the selection of the rotation factor. The complete OFDM signal can be generated after determining the best rotation factor set.
The improvement uses selected $q\%$ time-domain signal instead of all time-domain signal for optimization. The percentage $q\%$ of selection from time-domain signal directly affects the complexity of algorithm and the effect of PAPR reduction. After applying the above improvements, the O-IM-PTS algorithm scheme is shown in Fig. 5. The solid line part of the figure is the actual operation part. The dashed line part of the figure exists only in the theory, and the operation result is obtained from the solid line part through the equivalent transformation.
After using a combination of 4 improvement methods, the problem of high complexity due to the principle of the PTS algorithm has been greatly improved, resulting in the O-IM-PTS algorithm proposed in this paper. The amount of complexity reduced by the improvement schemes is shown in the Table 1.
Table 1. Complexity of the algorithm
The O-IM-PTS scheme is as follows:
- 1. The input constellation data ${X_z}[k ]$ are mapped into a sequence $X[k ]$ of length $2N$ by performing Hermitian symmetry after S/P conversion. The mapping relationship is as Eq. (21):
In Eq. (21), real(x) denotes the real part of the sequence x; image(x) denotes the imaginary part of the sequence x; and conj(x) denotes the complex conjugate sequence of the sequence x.
- 2. Perform sub-block partitioning on frequency domain sequence $X[k ]$ at $1 \le k \le N - 1$. It can be partitioned using any kind of methods, commonly used are adjacent partitioning, interleaved partitioning, pseudo-random partitioning. Let a total of M disjoint master sub-blocks be partitioned. Each sub-block is expanded to a sequence of length $2N$ by filling with zero. The set of elements selected by the $m$th master sub-block is denoted as $\{{{X_m}} \}$, and the sequence is denoted as ${X_m}[k ]$, $1 \le m \le M$.
- 3. Perform $2N$-point IDFT on all sequences ${X_m}[k ]$. The resulting time-domain sequences are ${x_m}[n ]$, $0 \le n \le 2N - 1$. The maximum possible power estimation is performed on time-domain sequence of all sub-blocks. Then select $q\%$ time-domain signal with larger maximum possible power for optimization. After the optimization, the rotation factor for $m$th sub-block is written as ${b_m}$. All the rotation factors are sent as the sideband information.
- 4. Each rotation factor obtained from optimization is multiplied with the time-domain sequence of the master sub-block. The optimization results of the slave sub-block are obtained by complex conjugate symmetry from results of the master sub-block. After the above results are summed, the time-domain signal, that is, the optimized OFDM signal, is obtained, which can be expressed as Eq. (22):
3. Experimental setup and results
A MATLAB simulation is performed to test the performance of different PTS algorithms. The simulation parameters are shown in Table 2. We choose a normal OFDM system with 485 subcarriers and 1024-point DFT. Each subcarrier uses 16-QAM modulation format. A total of 100000 OFDM symbols are used to generate Fig. 6. This parameter is set much larger than experiment parameter due to we want to simulate the difference of CCFD between different algorithms at 10−4 probability. In order to compare the performance of the algorithms under different sub-block number from 2 to 5, all PTS algorithms are using as PTS rotation factor scope. A Mersenne twister generator with the same random seed 178 is used to partite sub-blocks. The select ratio parameter $q\%$ of O-IM-PTS algorithms are set to 1.4%,4.2%,11.7% and 24.8% under sub-block number from 2 to 5 to balance performance and complexity.
Table 2. Simulation and experimental parameters
The simulation process is shown in Fig. 7. Random binary data is generated. Original OFDM algorithm, IM-PTS algorithms with four parameters sets and O-IM-PTS with four parameters sets are used to generate modulated signal from the same binary data. The PAPR and CCDF of different modulated signal are calculated to generate Fig. 6.
As can be seen from Fig. 6, the effect of PAPR reduction generally improves with the increase of the number of master sub-blocks. When the number of master sub-blocks is 2, only about 1.4% of the time-domain sequence is used to achieve almost the same effect as using the entire time-domain sequence. This ratio is about 4.2%, 11.7% and 24.8% when the number of master sub-blocks is 3, 4 and 5, and its effect of PAPR reduction is slightly inferior to the effect of optimization using the entire time domain sequence. The PAPR values at probabilities of ${10^0}$ to ${10^{ - 4}}$ are at most only about $0.03dB$ higher, which is acceptable.
The running time of MATLAB programs is easily affected by many factors, and the reference value is low, however, it is more intuitive. The running time of the O-IM-PTS algorithm is significantly reduced and the results are shown in Fig. 8. After reducing the time-domain sequence involved in the optimization, the MATLAB runtime for optimizing 100000 OFDM symbols is reduced to about 19%, 13%, 13%, and 39% of the original algorithm, respectively.
In order to compare the PTS algorithm proposed in this paper with algorithm used in literature [18], another simulation has been done with another set of parameters used in literature [18]. The set of parameters is shown in Table 3, and simulation results is shown in Fig. 9.
Table 3. Simulation parameters for comparison
As can be seen from Fig. 9, the effect of PAPR reduction generally improves with the increase of the number of master sub-blocks. The final simulation result in literature [18] using chaotic-SLM-PTS algorithms with 8 PTS subblocks is reducing PAPR of original signal from about 12.5 dB to 8 dB at probability 0.01. The simulation result in Fig. 9 shows that the PTS algorithms with 8 PTS subblocks proposed in the paper can reduce PAPR of original signal form about 12.5 dB to 7.8 dB at probability 0.01, which is slightly superior to the result in literature [18]. In terms of complexity, the proposed O-IM-PTS algorithms reduces the amount of computation of the SLM algorithm and chaotic systems used in the algorithm under the same PTS-related parameters. What’s more, compare to the original PTS algorithm, the complex of O-IM-PTS algorithms has been reduced too. In conclusion, the algorithm proposed in this paper achieves a slightly better result than the chaotic-SLM-PTS algorithms [18] with greatly reduced complexity.
An experiment is performed based on the first simulation with parameters in Table 2. We select the original signal, signal of IM-PTS algorithm with 5 sub-blocks, signal of O-IM-PTS algorithm with 5 sub-blocks to compare on the transmission system. The experiment parameters are shown in Table 2, and the experiment process is shown in Fig. 10. We select the first 500 OFDM symbols generate by simulation to reduce the amount of data collected by experiments. The cyclic prefix is added to reduce inter-symbol crosstalk, which used 1/4 length of the OFDM signal. Different kinds of OFDM data are spliced, so that the system can capture all signals at a time in order to control variables. The combined signal is transmitted by the experimental system. After transmission, the signal is divided into three groups. The cyclic prefix is removed and DFT operation is performed. After PTS demodulation, the EVM of different signal is compared. After the QAM De-mapping, the BER of different signal is compared.
The transmission system setup is shown in Fig. 11. Transmission system is an IMDD-OFDM system using the seven-core fiber. Transmit data generated by digital signal processing (DSP) algorithms is loaded into an arbitrary waveform generator (AWG) to generate electrical signals. The signal is amplified and then intensity-modulated using a Mach-Zehnder modulator with a light source to output the optical signal. The optical signal is amplified by an erbium-doped fiber amplifier (EDFA) and then divided into seven signals using an optical splitter, then the signal fans into the seven-core fiber for transmission. The length of the seven-core fiber is 2 km. The isolation parameters of the fiber core in the optical fiber are shown in the Table 4. It can be seen from the table that the isolation of different cores in the optical fiber is high enough, and the crosstalk between the cores should not become the main factor of noise at the receiving end. Therefore, the number of cores in the fiber does not affect the experimental results. After the fiber optic channel, the fiber is fanned out to seven cores. Since the purpose of the experiment is to compare the influence of signal PAPR on the receiving end, consider from the perspective of control variables, we take the signal of one of the cores. The signal is amplified by an EDFA, then attenuated by variable optical attenuator (VOA) to simulate the effect of different channel noise. Finally, a photodetector (PD) is used to convert the signal into an electrical signal, then a mixed signal oscilloscope (MSO) is used to capture the received signal.
Table 4. Seven core fiber isolation degree (dBm)
The total bit rate can be tantamount to the expression of (AWG sampling rate × entropy × subcarrier number /IFFT size/(1 + CP) ×number of core). According to the parameters shown in Table 2, the transmission rate is about ${10^{10}} \times 4 \times 485 \times 1500/({1024 \times 1500 \times 1.25} )\times 0.95 \times 7 = 100.8Gbit/s.\; $
In order to control channel parameters, we only compared different signals of the same core to measure the algorithm performance. The experimental results of BER of one core at different optical power are shown in Fig. 12. The BER at for the signal using PTS algorithm is not shown in the figure because no error occurs and the figure uses a logarithmic scale. It can be seen that in the case of high optical power, using the PTS algorithm helps to reduce the BER, and the difference between IM-PTS scheme and O-IM-PTS scheme can be regarded as experimental error. When the received optical power is low, the channel white noise becomes the main factor causing the BER, and whether or not the PTS algorithm is used has little improvement on the BER.
In order to reflect the effect of high PAPR on the signal in the case of high optical power and reduce the influence of occasional experimental error on the results in the case of low BER, we use the parameter: Error Vector Magnitude (EVM) to compare the received signal quality by comparing the constellation data. The EVM parameters are calculated from the QAM constellation diagram of each QAM symbol at the transmitter side and the QAM constellation diagram of each QAM symbol at the receiver side. The average value was calculated to represent the average effect of channel noise on a kind of signals at the current received optical power. The experimental results of average EVM of the same core used in Fig. 12 at different optical power are shown in Fig. 13.
It can be seen that EVM of all signals increases exponentially with increasing optical power. When the received optical power is $- 19dBm$ and, the transmission error is mainly caused by the noise in the channel, and there is little difference whether the PTS algorithm is used or not. When the received optical power is higher, as we can see from Fig. 13, the original signal has the highest EVM, and the results of the two PTS algorithms are similar.
Therefore, it can be concluded that when the nonlinear effect in the optical communication system becomes the main cause of signal noise, the use of the PTS algorithm can effectively improve the channel quality. In addition, it can also be seen from the experimental results that the performance of the O-IM-PTS algorithm is very close to that of the unoptimized algorithm with reducing a lot of complexity.
Figure 14 is the constellation of different signals at different received optical powers after equalization, and the results are similar to those of EVM. Due to the limited length of the optical fiber in the laboratory, the nonlinear effect in the optical fiber is not obvious in the experimental system. In addition, the lower optical attenuation of optical fiber means that there’s no need to use more optical amplifiers to compensate for optical signal attenuation. Therefore, in the experimental system, the signal PAPR has limited influence on the experimental results, which leads to the approximation of the experimental results between different signal.
4. Conclusion
In this paper we try to apply the PTS algorithm to the intensity-modulated OFDM system and we design the IM-PTS scheme, which theoretically ensures that the time-domain signal output by the algorithm is real value. What's more, we try to reduce the complexity of the IM-PTS algorithm from a total of four perspectives: limiting the optimized values, using equivalent operations, removing duplicate combinations, and using local optimization to fit the global optimization. In the case of M number of master sub-blocks, $2N$ number of DFT point, and four possible values of rotation factor, the four simplified methods when the PTS algorithm to generate each OFDM symbol reduce $2N \times 2M \times {4^M}$ times complex multiplications; ${4^M}/2$ times candidate signal generation and PAPR calculation; M times $2N$-point DFT; and when 2,3,4,5 master sub-blocks are used, the time-domain signal length for optimization is reduced to $1.4\%$, $4.2\%$, $11.7\%$, $24.8\%$, respectively. In the simulation, the MATLAB running time is reduced to about 19%,13%,13%,39% when optimizing 100000 OFDM symbols, respectively. The peak power of the OFDM signal generated by the O-IM-PTS scheme is only about $0.03dB$ higher at probability ${10^0}$ to ${10^{ - 4}}$, which is quite close to the effect of the IM-PTS scheme. Additional simulation with another set of parameters has been done in order to compare the proposed scheme with literature [18], the simulation results show that the algorithm proposed in this paper has a greater advantage in terms of computational complexity while achieving approximate results. Transmission experiments were conducted in a seven-core fiber IMDD-OFDM system at a rate of $100.8Gbit/s$. The BER and EVM of different signals at the receiver are compared under different received optical power. Both parameters reflect that when the received optical power is high, the reduced PAPR significantly improves the signal quality at the receiver, and when the received optical power is low, the white noise becomes the main factor affecting the received signal quality, whether or not to use the PTS algorithm does not have a significant impact. Comparing the two PTS algorithms, the experimental results are extremely close to each other which can be considered as experimental errors. The results show that the reduction of the complexity of the O-IM-PTS algorithm has a negligible impact on the signal quality.
Funding
National Key Research and Development Program of China (2018YFB1800901); National Natural Science Foundation of China (61835005, 62171227, 61727817, U2001601, 62035018, 61875248, 61935005, 61935011, 61720106015, 61975084); The Natural Science Foundation of the Jiangsu Higher Education Institutions of China (22KJB510031); Jiangsu Team of Innovation and Entrepreneurship; The Startup Foundation for Introducing Talent of NUIST.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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